How many families of subsets of $[n]$ are closed under intersection and union?

Question

Let $n \geq 1$. Is there a closed form, or an asymptotic estimate, for the number $g(n)$ of families $\mathcal{F} \subseteq 2^{[n]}$ which are closed under union and intersection?

An obvious upper bound is $2^{2^n}$. We can furnish a lower bound by considering chains $\varnothing=S_0 \subsetneq S_1 \subsetneq S_2 \cdots \subsetneq S_m=[n]$, and the combinatorics of counting such chains do not seem too daunting, and can be solved using the Partition function.

Can you calculate the answer for some small values of $n$, and then consult the Online Encyclopedia of Integer Sequences? — Gerry Myerson, Feb 21 '20 at 04:02
@GerryMyerson Thanks. I've done that, and it's A306445. Indeed there's a formula. — MathematicsStudent1122, Feb 21 '20 at 05:37
Good. I encourage you to write up a summary of what you've found there, and post it as an answer. — Gerry Myerson, Feb 21 '20 at 08:25

score 1 · Accepted Answer · answered Feb 23 '20 at 05:53

1

According to A306445, the formula is given by $$g(n)=1+\sum_{d=0}^{n}\sum_{i=d}^{n} {n \choose i}{i \choose i - d}f(d)$$

where $f(d)$ is the number of topologies on a finite set of $d$ elements. Per this, we have the estimate $\log_2 f(d) \sim d^2/4$.

answered Feb 23 '20 at 05:53

MathematicsStudent1122

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How many families of subsets of $[n]$ are closed under intersection and union?

1 Answers1